File:Ornstein-Uhlenbeck-5traces.svg

#! /usr/bin/env python3
# -*- coding:utf8 -*-

import matplotlib.pyplot as plt
from matplotlib.ticker import MultipleLocator
import numpy as np
from math import *

plt.rcParams['font.sans-serif'] = 'DejaVu Sans'
np.random.seed(42)

ntraces = 5
mu = 0. # long-term mean
tau = 1. # relaxation time scale
theta = 1. / tau # mean reversion rate
sigma = 1. # diffusion
D = sigma**2 / 2 # diffusion constant
sigmaX = sigma / sqrt(2 * theta) # standard deviation of X
t = np.linspace(0, 10, 1001)
dt = t[1:] - t[:-1]

fig = plt.figure(figsize=(520 / 90.0, 340 / 90.0), dpi=72)
plt.gca().axhspan(mu - sigmaX, mu + sigmaX, color='#e3e3e3')

for itrace in range(ntraces):
    randnorm = np.random.normal(0, 1, len(t))
    X = np.empty_like(t)
    X[0] = mu + sigmaX * randnorm[0]

    for i in range(1, len(t)):
        X[i] = X[i-1] + (mu - X[i-1]) * dt[i-1] * theta + sigma * sqrt(dt[i-1]) * randnorm[i]
    
    plt.plot(t, X, lw=1)

plt.grid(True)
plt.gca().yaxis.set_major_locator(MultipleLocator(1))
plt.xlim(t[0], t[-1])
plt.ylim(-2.2, 2.2)
plt.xlabel(r't / $\theta^{-1}$')
plt.ylabel('X')
plt.tight_layout()
plt.savefig('Ornstein-Uhlenbeck-5traces.svg')